Exercise 32, § 6

Let $(M_n)$, $(P_n)$ be two sequences of mutually disjoint finite sets (not all empty), indexed by the set $\mathbf{Z}$ of rational integers. Let $\alpha_n = \mathrm{Card}\ (M_n)$, $\beta_n = \mathrm{Card}\ (P_n)$. Suppose that there exists an integer $k > 0$ such that for each $n \in \mathbf{Z}$ and each integer $l \geqslant 1$ we have

Let M be the union of the family $(M_n)$ and let P be the union of the family $(P_n)$. Show that there exists a bijection $\varphi$ of M onto P such that

for each $n \in \mathbf{Z}$. (Consider a total ordering on each $\mathrm{M}_n$ (resp. $\mathrm{P}_n$) and take M (resp. P) to be the ordinal sum (§ 1, Exercise 3) of the family $(\mathrm{M}_n)_{n \in \mathbf{Z}}$ (resp. $(\mathrm{P}_n)_{n \in \mathbf{Z}}$). If $n_0$ is an index such that $\mathrm{M}_{n_0} \neq \emptyset$, consider the isomorphisms of M onto P which transform the least element of $\mathrm{M}_{n_0}$ into one of the elements of $\bigcup\limits_{j=n_0-k}^{n_0+k} \mathrm{P}_j$, and show that one of these isomorphisms satisfies the required conditions. Let $\delta$ be the least of the numbers

for all $n \in \mathbf{Z}$ and all $l \geqslant 1$. If $n \in \mathbf{Z}$ and $l \geqslant 1$ are such that, for example, $\beta_{n-k} + \beta_{n-k+1} + \cdots + \beta_{n+l+k} = \delta + \alpha_n + \alpha_{n+1} + \cdots + \alpha_{n+l}$, we may take $\varphi$ to be such that the least element of $\mathrm{P}_{n-k}$ is the image under $\varphi$ of the least element of $\mathrm{M}_n$.)